On certain commuting isometries, joint invariant subspaces and \(C^\ast \)-algebras (Q2220154)

An \(n\)-tuple \((V_1,\ldots,V_n)\) of commuting isometries on a Hilbert space is called a pure \(n\)-isometry if the operator product \(V_1\cdots V_n\) is a unilateral shift. As the first main result in the paper, the authors prove that any pure \(n\)-isometry is unitarily equivalent to a canonical model pure \(n\)-isometry. This provides a different, computational, but more explicit proof than that given by \textit{H. Bercovici} et al. [Acta Sci. Math. 72, No. 3--4, 639--661 (2006; Zbl 1164.47301)]. In the second main result, a characterization of joint invariant subspaces of model pure \(n\)-isometries is obtained. Operator-valued inner functions play an essential role in such a characterization. Lastly, the authors are concerned with \(C^{*}\)-algebras generated by multiplication operators by coordinate functions restricted to an invariant subspace in the Hardy space \(H^2(\mathbb{D}^n)\) over the unit polydisk. More specifically, consider the \(n\)-tuple of commuting isometries \((M_{z_1},\ldots,M_{z_n})\) on \(H^2(\mathbb{D}^n)\). For any closed subspace \(\mathcal{S}\) invariant for all \(M_{z_1},\ldots, M_{z_n}\), let \(\mathcal{T}(\mathcal{S})\) denote the \(C^{*}\)-algebra generated by \(\{M_{z_1}|_{\mathcal{S}},\ldots, M_{z_n}|_{\mathcal{S}}\}\). It is shown that whenever \(\mathcal{S}\) and \(\mathcal{S}'\) are invariant subspaces of finite codimension, \(\mathcal{T}(\mathcal{S})\) and \(\mathcal{T}(\mathcal{S}')\) are isomorphic as \(C^{*}\)-algebras. This extends a result of \textit{M. Seto} [Can. Math. Bull. 47, No. 3, 456--467 (2004; Zbl 1088.47023)] from \(n=2\) to arbitrary \(n\). For the entire collection see [Zbl 1455.47001].